Problem of the Week

Problem 7

In this problem, an $n$ -list is a sequence of $n$ non-negative numbers, where $n$ is a positive integer. A list is an $n$ -list for some positive integer $n$ .

The arithmetic mean $\alpha$ of the $n$ -list $x_1, ..., x_n$ is given by $\alpha = (x_1 + ... + x_n)/n$ ; the geometric mean $\gamma$ is given by $\gamma = (x_1 ... x_n)^{1/n}$ .

(i) Show that $\alpha \ge \gamma$ for all $2$ -lists.

(ii) Show that if $\alpha \ge \gamma$ for all $k$ -lists and all $l$ -lists, then $\alpha \ge \gamma$ for all $kl$ -lists.

(iii) Show that if $\alpha \ge \gamma$ for all $(k+1)$ -lists, then $\alpha \ge \gamma$ for all $k$ -lists.

(iv) Deduce that $\alpha \ge \gamma$ for all lists.

Submit a solution!

Name

Email

School / Institution

Details

Upload solution (*.jpg, *.pdf, *.docx, *.png accepted, max 2mb)

Publish consent

You can display my name on the CMS website if I successfully solve the problem.

Data Protection

I consent to providing this information to Cambridge Maths School. It will be processed in accordance with the Trust Privacy Policy.